How far a wave can remember

Michelson’s interferometer splits a beam in two, sends the halves down arms of different length, and recombines them. With a laser the fringes are there whatever the difference in arm length. With a filament lamp they appear only when the arms are matched to within a few micrometres, and vanish if one mirror is moved by the thickness of a cigarette paper. The instrument is the same instrument. What has changed is the source’s memory of its own phase.

Two beams, one source, one delay #

Send a wave down two paths differing by $\Delta$ and add the results. If the field at the source is $E(t)$, the recombined intensity averaged over a detection time is

$$I(\Delta) \propto 2\langle |E|^2\rangle + 2\,\mathrm{Re}\,\big\langle E^*(t)\,E(t + \Delta/c)\big\rangle,$$

and the second term is the autocorrelation of the field with itself, delayed. If the source is perfectly monochromatic, $E \propto e^{-i\omega t}$, the correlation is $e^{i\omega\Delta/c}$ and has modulus one for every delay: the fringes never fade.

The idealisation the coherence length measures the failure of is a sinusoid of one frequency and infinite extent, whose value at any instant fixes its value at every other. Such a source would give fringes at any path difference whatever, including a path difference of a kilometre, because there is nothing in it that could get out of step with anything else. No source is like this, and the reason is not that real sources are imperfect: a wave of exactly one frequency has to have been going on for ever.

Real sources are not monochromatic. A spectral line has a width because the emitting atoms are moving, colliding, and radiating for a finite time, so the field is a train of wave packets with uncorrelated phases rather than an endless sinusoid. Correlating such a field with a delayed copy of itself gives one when the delay is short compared with a packet and nothing when it is long — and this is where the arrow that distinguishes a spreading disturbance from a gathering one enters an argument that otherwise contains no time direction at all.

What an atom actually emits is a packet — a carrier with an envelope, lasting as long as the emission does. Correlate one of these with a copy of itself delayed by less than the envelope’s width and the two overlap; delay it by more and there is nothing left to correlate against. A real source emits a stream of such packets with random phases between them, so the correlation is decided entirely by the individual packet and not at all by the stream.

The Wiener–Khinchin theorem turns that into an exact statement. The autocorrelation of a stationary signal and its power spectrum are a Fourier transform pair, so

$$\gamma(\tau) = \frac{\int S(\nu)\,e^{-2\pi i \nu\tau}\,d\nu}{\int S(\nu)\,d\nu}, \qquad \tau = \Delta/c,$$

and the fringe visibility is $|\gamma|$. The spectrum and the visibility are the same information, written two ways.

The convention, and the three things it rounds together #

The number quoted for a source is

$$L_c = \frac{\lambda^2}{\Delta\lambda},$$

which is $c$ divided by the bandwidth in frequency. It is a scale rather than a distance at which anything happens, and the three shapes in the opening figure show how far apart the actual behaviours are.

A flat band — an ideal filter with sharp edges — transforms to a sinc, so the visibility falls, reaches zero, and comes back up. There really are path differences at which such a source gives no fringes and larger ones at which it gives some again. A Gaussian line transforms to a Gaussian and decays monotonically with no revival, which is the well-behaved case and the one most nearly true of Doppler-broadened lines. A Lorentzian — the shape a radiative lifetime produces — transforms to a decaying exponential, which falls fast at first and then keeps a long tail: at four times $L_c$ a Lorentzian still shows a few per cent of visibility while a Gaussian shows none.

Two lines, and a visibility that comes back #

The clearest demonstration that fading is disagreement between components rather than decay is a source with only two of them.

The sodium D lines sit at 589.00 and 589.59 nm, 0.597 nm apart. Each produces its own fringe pattern; the two patterns have slightly different spacings, so as the path difference grows they slide relative to one another. When they are half a fringe out of step the maxima of one fall on the minima of the other and the fringes vanish completely; when they are a whole fringe out of step the patterns coincide again and the fringes come back at full strength.

The same arithmetic appears in one dimension at a frequency an ear can hear. Two tones close together produce an amplitude that waxes and wanes at their difference frequency, and the sodium doublet’s revivals are that envelope plotted against path difference instead of against time. Nothing decays in either picture. What changes is whether the two components agree, and agreement is periodic when there are only two of them — which is why the visibility comes back rather than dying away.

Michelson’s method is worth appreciating for what it does not require. There is no grating, no prism, no dispersion of any kind: the spectrum is recovered by moving a mirror and counting. Take the visibility as a function of mirror position, transform it, and the spectrum comes out — which is Fourier transform spectroscopy, and it is how infrared spectra are taken today, because at long wavelengths a moving mirror is far easier to build than a good grating.

A short coherence length is a ruler #

Everything so far treats a short coherence length as a limitation. Turn the argument over and it becomes the most useful thing about a broad source: fringes appear only where the two arms are matched, so the mirror position at which they appear is a measurement of the other arm’s length, and the sharpness of that measurement is the coherence length itself.

Point one arm at a layered object instead of a mirror. Every interface in it returns some light, each with its own delay, and the reference mirror sees fringes only from the one interface whose delay it currently matches. Scanning the mirror therefore reads out a depth profile, one interface at a time, with a depth resolution equal to the width of the visibility envelope. That is optical coherence tomography, and it is why it is done with the broadest source available rather than with a laser: a laser’s kilometre of coherence would return every interface at once and resolve nothing.

The number in every such instrument’s specification is the one the opening figure computes. For a Gaussian line the visibility halves at 0.44 of $\lambda^2/\Delta\lambda$, so a source at 840 nm with a 50 nm bandwidth gives about 6 µm of depth resolution — enough to see the layers of a retina. The constant 0.44 is not a rule of thumb: it is the width of the transform of a Gaussian, and it is the same 0.44 the figure returns from summing over the spectrum. The distinction the essay’s second refutation makes, between a convention and a distance at which something happens, is the difference between quoting $\lambda^2/\Delta\lambda$ and quoting what an instrument actually resolves.

It also explains why the flat-band curve would be a poor choice for such an instrument even though its envelope is narrowest at first. Its revivals are ghost interfaces — a reflection appearing at a depth where there is nothing — and an ideal filter with sharp edges is exactly the wrong thing to put in front of the lamp.

What a mirror’s travel buys #

The same inversion applies to the spectrometer built out of the same apparatus, and it turns the question round: not how far the mirror may move before the fringes die, but how far it must move before the spectrum is resolved.

The visibility is the transform of the spectrum, and a transform truncated at some maximum delay cannot resolve structure finer than the reciprocal of that delay. Move the mirror by a centimetre, and since the light traverses the arm twice the largest path difference is two centimetres, giving a resolution of half a reciprocal centimetre — which is the specification of an ordinary benchtop infrared spectrometer, and it comes from a length rather than from any optical quality. Wanting ten times finer means moving the mirror ten times further, and nothing else.

Two things make that trade worth taking in the infrared, and neither is obvious. Every detected photon contributes to every point of the recovered spectrum, because the measurement is a sum over the whole band at each mirror position rather than a series of measurements of narrow slices — so a spectrum of $N$ resolution elements is acquired with the noise of one long measurement rather than of $N$ short ones. And there is no slit: the aperture can be a large round hole rather than a narrow line, which admits far more light than a grating instrument of the same resolution. Both advantages matter most where detectors are noisy, which is precisely the infrared.

What the numbers are #

The range is enormous and it is worth having laid out.

Sunlight across the visible band, $\Delta\lambda \approx 300$ nm at $\lambda = 550$ nm, gives $L_c \approx 1\,\mathrm{\mu m}$ — about two wavelengths. That is why white-light fringes show only a handful of orders and why the central fringe is identifiable at all: it is the only one at which every colour agrees, and it is the one used to find zero path difference in an interferometer. It is also why a thin film shows colours rather than a fringe count — a film thicker than a micrometre has passed the point where the orders can be told apart.

A filament lamp behind a 10 nm filter gives 30 µm. A low-pressure sodium lamp, whose lines are narrow, gives a few centimetres. A helium–neon laser running on many longitudinal modes gives perhaps 20 cm; a single-mode stabilised one gives hundreds of metres, and the best gives kilometres.

A hot body is the worst possible source for an interferometer, and the spectrum says why. It is not a line at all but a broad continuum a thousand nanometres wide, so its coherence length is a wavelength or two and its fringes exist only where the two arms are matched to within a wavelength. Every improvement in coherence — a filter, a discharge lamp, a laser — is a narrowing of that curve, and nothing else.

There is another kind of coherence, and it is worth naming so that this essay’s silence about it is deliberate. Fringes in a two-slit experiment require the light reaching the two slits to agree in phase across the beam, which is spatial coherence and is a question about the size of the source rather than about its bandwidth. Both must hold for fringes to appear; they are independent; and the path difference in a Michelson tests only the temporal one.

The two coherences, and the one this essay is about #

A source can be spectrally pure and spatially incoherent, or the reverse. The distinction is worth stating because the two are constantly conflated.

Temporal coherence is the question this essay asks: over what delay does the field agree with itself? The answer is $c/\Delta\nu$ and is fixed by the spectrum.

Spatial coherence asks over what transverse separation the field agrees with itself, and the answer is fixed by the angular size of the source rather than by its colour. A large source lights two slits with fields that have no fixed phase relation, and no filter narrow enough to give a coherence length of metres will make fringes appear if the source subtends too large an angle. Both are conditions on the sum, not on the individual waves, which is why neither can be repaired by making the light brighter.

The two are Fourier transforms of different things — the spectrum in one case, the source’s brightness distribution across the sky in the other — and the second is the basis of stellar interferometry, which is not this site’s to write about. What matters here is that a fading of fringes has two possible causes and that changing the path difference tests only one of them.

The linewidth as a clock #

Reading the argument backwards turns the coherence length into a statement about time, and the statement is sharper than the optics suggests.

An atom radiating for a time $\tau_0$ before something interrupts it — a collision, or spontaneous emission running its course — produces a wave train of length $c\tau_0$. Transforming a truncated sinusoid gives a Lorentzian line of width $\Delta\nu = 1/2\pi\tau_0$, so the linewidth is the interruption rate, expressed in hertz. Sodium’s excited state lives 16 ns, which gives a natural linewidth of about 10 MHz and a natural coherence length of about 5 m.

Measured sodium lines are far broader than that — around 1.5 GHz in a hot vapour — and the excess is Doppler broadening: the atoms are moving, each carrying its own shift, so the observed line is the natural Lorentzian smeared by the distribution of speeds in the vapour. That smearing is a Gaussian, which is why real spectral lines are neither Lorentzian nor Gaussian but the convolution of the two, and why their visibility curves have a fast Gaussian fall with a slow Lorentzian tail underneath.

The linewidth is itself a thermometer. Each atom’s emission is shifted by its own velocity along the line of sight, so the spread of speeds in the vapour becomes a spread of frequencies and the observed line is broader than any single atom’s. Cooling the vapour narrows the distribution and lengthens the coherence, which is one of the several reasons a spectroscopist cools things.

So the fading of fringes in an interferometer is a measurement of how long the atoms in the source go undisturbed. The mirror is a stopwatch with a range of nanoseconds and a resolution of femtoseconds, and it is reading the lifetime of an excited state from a distance in millimetres.

Visibility is not only about coherence #

One honest qualification to the claim that the apparatus appears nowhere. The measured visibility is the product of two factors:

$$V = \frac{2\sqrt{I_1 I_2}}{I_1 + I_2}\,|\gamma(\tau)|,$$

and only the second is the source’s. The first is an inequality between the two beams, and it falls off very gently — a beamsplitter that sends 60 per cent one way and 40 per cent the other still gives 0.98 — which is why the coherence factor dominates any real measurement and why the refutation above stands. But it is not identically one, and a visibility of 0.9 measured on a source expected to give 1.0 is far more often an unequal splitter, a misaligned mirror or a detector seeing stray light than it is a spectrum with unexpected wings.

The practical consequence is that the shape of a visibility curve is trustworthy and its absolute height is not. Normalising the measured curve to its value at zero path difference removes every instrumental factor that does not depend on the delay, which is all of the ones listed above, and leaves $|\gamma|$ — which is why a Fourier transform spectrometer is calibrated at zero delay and why the interferogram’s central spike is the most carefully measured point in the scan.

The measurement that fixed the metre #

Michelson’s use of this is worth stating because the limit he was working against is the subject of this essay. In 1892 the metre was a scratched bar in a vault, and the proposal was to define it instead as a number of wavelengths of a chosen spectral line. Doing that requires counting fringes over half a metre, which requires a source whose fringes survive half a metre of path difference — so the choice of line was a choice of coherence length, and the red line of cadmium was picked over every brighter alternative because it was the narrowest available.

Even so it would not reach directly, and the answer was a chain of nine etalons each roughly twice the length of the last, every one compared against its neighbour over a path difference short enough for the fringes to hold. The result, 1,553,163.5 wavelengths to the metre, stood for decades. What limited the experiment was not the optics, the mechanics or the counting: it was how long a cadmium atom radiates undisturbed.

Where the model stops #

Stationarity. The whole treatment assumes the statistics of the source do not change during the measurement. A pulsed source, or one being tuned, has a visibility that depends on when it was looked at, and the transform relation does not apply.

One polarisation. Two beams of orthogonal polarisation do not interfere at all whatever their coherence, so a source with a randomly varying polarisation loses visibility for a reason that has nothing to do with its spectrum. Coherence in the full sense is a matrix, and this essay uses one element of it.

Linear, dispersionless arms. If the two arms contain different thicknesses of glass, each wavelength acquires a different extra delay and the visibility falls for a reason belonging to the instrument after all. That is why a Michelson has a compensating plate: not to equalise the path but to equalise the dispersion, so that the fading measured is the source’s.

What the pictures cannot show #

Every curve here is a modulus. The transform of the spectrum is complex, and its phase carries the position of the line centre; taking the modulus throws that away, which is exactly why visibility alone cannot distinguish a line at 589.0 nm from one at 589.6 nm, and why the full transform is needed to recover a spectrum rather than merely a linewidth.

Nothing here shows the fringes themselves. Visibility is a summary — $(I_{\max} - I_{\min})/(I_{\max} + I_{\min})$ — extracted from a pattern that the figures never draw, and a visibility of 0.5 does not say whether the fringes are half as deep or half as many.

And the doublet curve returns to exactly one for ever, because the model has two lines of zero width. Real sodium lines have widths, so the revivals decay, and the envelope of the revivals is the transform of a single line’s shape. Both effects are present in a real measurement and the figure separates them by removing one.

Where the ladder goes next #

The rung below established that two independent lamps never make fringes. This one asks what a single source can be made to interfere with — itself, delayed — and finds that the answer is a Fourier transform of its spectrum.

The rungs above are where that transform is used rather than illustrated: interferometry as a spectrometer, where visibility against mirror position is measured and inverted; and coherence as a resource, where the question is how many independent things a source can be split into before the fringes go, which turns into the counting argument behind how many modes a cavity has.

What is worth carrying is the shape of the argument rather than the result. A quantity was measured as a function of a delay, and it turned out to be the transform of a quantity measured as a function of a frequency. Whenever an experiment sweeps a delay, it is very often measuring a spectrum without meaning to.